Solve the following paris of linear (simultaneous) equation using method of elimination by substution:
`(x)/(6) + (y)/(15) = 4`
`(x)/(3) - (y)/(12) = 4(3)/(4)`

Answer

To solve the given pair of linear equations using the method of elimination by substitution, we will follow these steps:

**Given Equations:**
1. \(\frac{x}{6} + \frac{y}{15} = 4\)
2. \(\frac{x}{3} - \frac{y}{12} = 4 \cdot \frac{3}{4}\)

### Step 1: Convert the equations to a simpler form
First, we will eliminate the fractions by multiplying through by the least common multiple (LCM) of the denominators.

**For the first equation:**
The LCM of 6 and 15 is 30. Multiply the entire equation by 30:
\[ 30 \left(\frac{x}{6}\right) + 30 \left(\frac{y}{15}\right) = 30 \cdot 4 \]
This simplifies to:
\[ 5x + 2y = 120 \quad \text{(Equation 1)} \]

**For the second equation:**
The LCM of 3 and 12 is 12. Multiply the entire equation by 12:
\[ 12 \left(\frac{x}{3}\right) - 12 \left(\frac{y}{12}\right) = 12 \cdot 4 \cdot \frac{3}{4} \]
This simplifies to:
\[ 4x - y = 36 \quad \text{(Equation 2)} \]

### Step 2: Express one variable in terms of the other
From Equation 2, we can express \(y\) in terms of \(x\):
\[ y = 4x - 36 \]

### Step 3: Substitute into the first equation
Now, we will substitute \(y\) in Equation 1:
\[ 5x + 2(4x - 36) = 120 \]
Expanding this gives:
\[ 5x + 8x - 72 = 120 \]
Combine like terms:
\[ 13x - 72 = 120 \]

### Step 4: Solve for \(x\)
Add 72 to both sides:
\[ 13x = 192 \]
Now, divide by 13:
\[ x = \frac{192}{13} \approx 14.77 \]

### Step 5: Substitute \(x\) back to find \(y\)
Now substitute \(x\) back into the equation for \(y\):
\[ y = 4\left(\frac{192}{13}\right) - 36 \]
Calculating this gives:
\[ y = \frac{768}{13} - 36 \]
Convert 36 to a fraction with a denominator of 13:
\[ y = \frac{768}{13} - \frac{468}{13} = \frac{300}{13} \approx 23.08 \]

### Final Answer
Thus, the solution to the system of equations is:
\[ x = \frac{192}{13}, \quad y = \frac{300}{13} \]

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